Idea

In the general context of cohomology, as described there, a cocycle representing a cohomology class on an object XX with coefficients in an object AA is a morphismc:X→Ac : X \to A in a given ambient (∞,1)-toposH\mathbf{H}.

The same applies with the object AA taken as the domain object: for BB yet another object, the BB-valued cohomology of AA is similarly H(A,B)=π0H(A,B)H(A,B) = \pi_0 \mathbf{H}(A,B). For [k]∈H(A,B)[k] \in H(A,B) any cohomology class in there, we obtain an ∞-functor

[k(−)]:H(X,A)→H(X,B)
[k(-)] : \mathbf{H}(X,A) \to H(X,B)

from the AA-valued cohomology of XX to its BB-valued cohomology, simply from the composition operation

In practice one is interested in this notion for particularly simple objects BB, notably for BB an Eilenberg-MacLane objectBnK\mathbf{B}^n K for some component KK of a spectrum object. This serves to characterize cohomology with coefficients in a complicated object AA by a collection of cohomology classes with simpler coefficients. Historically the name characteristic class came a little different way about, however (see also historical note on characteristic classes).

Then with the usual notation Hn(X,K):=H(X,BnK)H^n(X,K) := H(X, \mathbf{B}^n K) a given characteristic class in degree nn assigns

[k(−)]:H(X,A)→Hn(X,K).
[k(-)] : \mathbf{H}(X,A) \to H^n(X,K)
\,.

Moreover, recall from the discussion at cohomology that to every cocyclec:X→Ac : X \to A is associated the object P→XP \to X that it classifies – its homotopy fiber – which may be thought of as an AA-principal ∞-bundle over XX with classifying map X→AX \to A. One typically thinks of the characteristic class [k(c)][k(c)] as characterizing this principal ∞-bundlePP.

kk-Invariants

Level in ∞\infty-Chern-Simons theory

Of Lagrangian submanifolds

Classes in the sense of Fuks

In (Fuks (1987), section 7) an axiomatization of characteristic classes is proposed. We review the definition and discuss how it is a special case of the one given above.

Fuks’s definition

Fuks considers a base category 𝒯\mathcal{T} of “spaces” and a category 𝒮\mathcal{S} of spaces with a structure (for example, space together with a vector bundle on it), this category should be a category over 𝒯\mathcal{T}, i.e. at least equipped with a functorU:𝒮→𝒯U : \mathcal{S}\to\mathcal{T}.

A morphism of categories with structures is a morphism in the overcategoryCat/𝒯/\mathcal{T}, i.e. a morphism U→U′U\to U' is a functor F:dom(U)→dom(U′)F: dom(U)\to dom(U') such that U′F=UU' F = U.

Suppose now the category 𝒯\mathcal{T} is equipped with a cohomology theory which is, for purposes of this definition, a functor of the form H:𝒯op→AH : \mathcal{T}^{op} \to A where AA is some concrete category, typically category of T-algebras for some algebraic theory in Set, e.g. the category of abelian groups. Define ℋ=ℋH\mathcal{H} = \mathcal{H}_H as a category whose objects are pairs (X,a)(X,a) where XX is a space (= object in 𝒯\mathcal{T}) and a∈H(X)a\in H(X). This makes sense as AA is a concrete category. A morphism (X,a)→(Y,b)(X,a)\to (Y,b) is a morphism f:X→Yf: X\to Y such that H(f)(b)=aH(f)(b) = a. We also denote f*=H(f)f^* = H(f), hence f*(b)=af^*(b) = a.

A characteristic class of structures of type 𝒮\mathcal{S} with values inHH in the sense of (Fuks) is a morphism of structures h:𝒮→ℋHh: \mathcal{S}\to\mathcal{H}_H over 𝒯\mathcal{T}. In other words, to each structure SS of the type 𝒮\mathcal{S} over a space XX in 𝒯\mathcal{T} it assigns an element h(S)h(S) in H(X)H(X) such that for a morphism t:S→Tt: S\to T in 𝒮\mathcal{S} the homomorphism (U(t))*:H(Y)→H(X)(U(t))^* : H(Y)\to H(X), where Y=U(T)Y = U(T), sends h(S)h(S) to h(T)h(T).